Topologically twisted index of $T[SU(N)]$ at large $N$

We compute, in the large $N$ limit, the topologically twisted index of the 3d $T[SU(N)]$ theory, namely the partition function on $\Sigma_{\mathfrak{g}} \times S^1$, with a topological twist on the Riemann surface $\Sigma_{\mathfrak{g}}$. To provide an expression for this quantity, we take advantage of some recent results obtained for five dimensional quiver gauge theories. In case of a universal twist, we correctly reproduce the entropy of the universal black hole that can be embedded in the holographically dual solution.